Measures of Variability

0

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0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

Variability

Measure of Variability (Dispersion, Spread)

• Variance, standard deviation

• Range

• Inter-Quartile Range

• Pseudo-standard deviation

Range

Range

Definition

Let min = the smallest observation

Let max = the largest observation

Then Range =max - min

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Range

Inter-Quartile Range (IQR)

Inter-Quartile Range (IQR)

Definition

Let Q1 = the first quartile,

Q3 = the third quartile

Then the

Inter-Quartile Range

= IQR = Q3 - Q1

0

0.02

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0 5 10 15 20 25Q1 Q3

25% 25%

50%

Inter-Quartile Range

Example

The data Verbal IQ on n = 23 students arranged in increasing order is:

80 82 84 86 86 89 90 94

94 95 95 96 99 99 102 102

104 105 105 109 111 118 119

Example

The data Verbal IQ on n = 23 students arranged in increasing order is:

80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 102 104 105 105 109 111 118 119

Q2 = 96Q1 = 89 Q3 = 105min = 80 max = 119

Range

Range = max – min = 119 – 80 = 39

Inter-Quartile Range

= IQR = Q3 - Q1 = 105 – 89 = 16

Some Comments

• Range and Inter-quartile range are relatively easy to compute.

• Range slightly easier to compute than the Inter-quartile range.

• Range is very sensitive to outliers (extreme observations)

Varianceand

Standard deviation

Sample Variance

Let x1, x2, x3, … xn denote a set of n numbers.

Recall the mean of the n numbers is defined as:

n

xxxxx

n

xx nn

n

ii

13211

The numbers

are called deviations from the the mean

xxd 11

xxd 22

xxd 33

xxd nn

The sum

is called the sum of squares of deviations from the the mean.

Writing it out in full:

or

n

ii

n

ii xxd

1

2

1

2

223

22

21 ndddd

222

21 xxxxxx n

The Sample Variance

Is defined as the quantity:

and is denoted by the symbol

111

2

1

2

n

xx

n

dn

ii

n

ii

2s

Example

Let x1, x2, x3, x3 , x4, x5 denote a set of 5 denote the set of numbers in the following table.

i 1 2 3 4 5

xi 10 15 21 7 13

Then

= x1 + x2 + x3 + x4 + x5

= 10 + 15 + 21 + 7 + 13

= 66

and

5

1iix

n

xxxxx

n

xx nn

n

ii

13211

2.135

66

The deviations from the mean d1, d2, d3, d4, d5 are given in the following table.

i 1 2 3 4 5

xi 10 15 21 7 13

di -3.2 1.8 7.8 -6.2 -0.2

The sum

and

n

ii

n

ii xxd

1

2

1

2

22222 2.02.68.78.12.3

80.112

04.044.3884.6024.324.10

2.28

4

8.112

11

2

2

n

xxs

n

ii

The Sample Standard Deviation s

Definition: The Sample Standard Deviation is defined by:

Hence the Sample Standard Deviation, s, is the square root of the sample variance.

111

2

1

2

n

xx

n

ds

n

ii

n

ii

In the last example

31.52.28

4

8.112

11

2

2

n

xxss

n

ii

Interpretations of s

• In Normal distributions– Approximately 2/3 of the observations will lie

within one standard deviation of the mean– Approximately 95% of the observations lie

within two standard deviations of the mean– In a histogram of the Normal distribution, the

standard deviation is approximately the distance from the mode to the inflection point

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s

Inflection point

Mode

s

2/3

s

2s

Example

A researcher collected data on 1500 males aged 60-65.

The variable measured was cholesterol and blood pressure.

– The mean blood pressure was 155 with a standard deviation of 12.

– The mean cholesterol level was 230 with a standard deviation of 15

– In both cases the data was normally distributed

Interpretation of these numbers

• Blood pressure levels vary about the value 155 in males aged 60-65.

• Cholesterol levels vary about the value 230 in males aged 60-65.

• 2/3 of males aged 60-65 have blood pressure within 12 of 155. Ii.e. between 155-12 =143 and 155+12 = 167.

• 2/3 of males aged 60-65 have Cholesterol within 15 of 230. i.e. between 230-15 =215 and 230+15 = 245.

• 95% of males aged 60-65 have blood pressure within 2(12) = 24 of 155. Ii.e. between 155-24 =131 and 155+24 = 179.

• 95% of males aged 60-65 have Cholesterol within 2(15) = 30 of 230. i.e. between 230-30 =200 and 230+30 = 260.

Measures of Variability

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

Variability

Measure of Variability (Dispersion, Spread)

• Variance, standard deviation

• Range

• Inter-Quartile Range

• Pseudo-standard deviation

Range

Range =max – min

Interquartile range (IQR)

IQR = Q3 – Q1

The Sample Variance

111

2

1

2

2

n

xx

n

ds

n

ii

n

ii

2s

The Sample standard deviation

111

2

1

2

n

xx

n

ds

n

ii

n

ii

2s

A Computing formula for:

Sum of squares of deviations from the the mean :

The difficulty with this formula is that will have many decimals.

The result will be that each term in the above sum will also have many decimals.

n

ii xx

1

2

x

The sum of squares of deviations from the the mean can also be computed using the following identity:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

To use this identity we need to compute:

and 211

n

n

ii xxxx

222

21

1

2n

n

ii xxxx

Then:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

11 and

2

1

1

2

1

2

2

nn

x

x

n

xxs

n

iin

ii

n

ii

11

and

2

1

1

2

1

2

nn

x

x

n

xxs

n

iin

ii

n

ii

Example

The data Verbal IQ on n = 23 students arranged in increasing order is:

80 82 84 86 86 89 90 94

94 95 95 96 99 99 102 102

104 105 105 109 111 118 119

= 80 + 82 + 84 + 86 + 86 + 89

+ 90 + 94 + 94 + 95 + 95 + 96 + 99 + 99 + 102 + 102 + 104

+ 105 + 105 + 109 + 111 + 118 + 119 = 2244

= 802 + 822 + 842 + 862 + 862 + 892

+ 902 + 942 + 942 + 952 + 952 + 962 + 992 + 992 + 1022 + 1022 + 1042

+ 1052 + 1052 + 1092 + 1112

+ 1182 + 1192 = 221494

n

iix

1

n

iix

1

2

Then:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

652.2557

23

2244221494

2

11 and

2

1

1

2

1

2

2

nn

x

x

n

xxs

n

iin

ii

n

ii

26.116

22

652.2557

2223

2244221494

2

11 Also

2

1

1

2

1

2

nn

x

x

n

xxs

n

iin

ii

n

ii

26.116

22

652.2557

2223

2244221494

2

782.10

A quick (rough) calculation of s

The reason for this is that approximately all (95%) of the observations are between

and

Thus

4

Ranges

sx 2.2sx

sx 2max .2min and sx .22minmax and sxsxRange

s4

4

Range Hence s

Example

Verbal IQ on n = 23 students min = 80 and max = 119

This compares with the exact value of s which is 10.782.The rough method is useful for checking your calculation of s.

75.94

39

4

80-119s

The Pseudo Standard Deviation (PSD)

The Pseudo Standard Deviation (PSD)

Definition: The Pseudo Standard Deviation (PSD) is defined by:

35.1

Range ileInterQuart

35.1

IQRPSD

Properties

• For Normal distributions the magnitude of the pseudo standard deviation (PSD) and the standard deviation (s) will be approximately the same value

• For leptokurtic distributions the standard deviation (s) will be larger than the pseudo standard deviation (PSD)

• For platykurtic distributions the standard deviation (s) will be smaller than the pseudo standard deviation (PSD)

Example

Verbal IQ on n = 23 students Inter-Quartile Range

= IQR = Q3 - Q1 = 105 – 89 = 16

Pseudo standard deviation

This compares with the standard deviation

85.1135.1

16

35.1

IQRPSD

782.10s

Summary